The construction of periodic functions

Problem

For each requirement, find a corresponding term of a periodic function defined in \(\mathbb{R}\).

1. The graph of the function \(g\) is obtained from the graph of the function \(x\mapsto \sin(x)\) defined in \(\mathbb{R}\) by reflection at the \(y\)-axis.

2. The function \(h\) has the codomain \([1;3]\).

3. The function \(k\) possesses the period \(\pi\).

Solution of part a

If a function \(g\) is to be obtained by reflection of the function \(f\) at the \(y\)-axis, we have \(g(x)=f(-x)\). In our case, we obtain \(g(x)=\sin(-x)\). In view of the point symmetry of the sine function with respect to the origin the function \(g\) is of the form \(g(x)=-\sin(x)\).

We can easily check this result with the help of Sage by plotting in one graph the sine function (blue) and its reflection (red).

Solution of part b

In this part, we again make use of the sine function. The codomain of the sine function with amplitude \(1\) is given by \([-1;1]\). In order to obtain the required codomain, we can shift the sine function together with its codomain by a constant of \(2\) towards the top and thus obtain \(h(x)=\sin(x)+2\).

We check the codomain of the function by means of Sage:

Solution of part c

The period of the sine function can be adjusted by means of a parameter \(a\) in front of the function’s argument. Our function will thus be of the form \(k(x)=\sin(ax)\). The period for \(a=1\) is \(2\pi\). The requested period is only half as long so that the parameter \(a\) needs to be doubled. We thus obtain \(k(x)=\sin(2x)\).

The period of the function can be checked by means of Sage. In the plot, one period is represented in red.